collinear 79

closure property A

BINARY OPERATION

on a set Sis

said to be closed if the combination of two elements in

that set yields another member of that set. For exam-

ple, the set of positive whole numbers is closed under

addition, since the sum of any two positive integers is

itself a positive integer. This set is also closed under

multiplication (the product of two positive whole num-

bers is a positive integer), but not subtraction: if nand

mare positive whole numbers, then n – m could be

zero or negative and thus no longer in the set of posi-

tive whole numbers. (For instance, 3 – 7 = –4.)

For a more unusual example, consider the set Sof all

whole numbers that can be expressed as the sum of two

SQUARE NUMBERS

. We have S= {0, 1, 2, 4, 5, 8, 9, 10, 13,

16, 17, 20, 25,…}. (For instance, 0 = 02+ 02, 5 = 12+ 22,

20 = 22+ 42, and 25 = 02+ 52= 32+ 42.) Surprisingly,

this set is closed under multiplication. For example, both

5 and 8 belong to S, and so does 5 ×8 = 40. (We have

40 = 22+ 62.) Also, 10 and 13 belong to S, and so too

does 130. (We have 130 = 32+ 112.) This general

observation follows from the algebraic identity that if

N=a2+b2and M= c2+ d2; then N×M= (ac + bd)2+

(ad – bc)2.

This set Sis also closed under exponentiation: if N

and Mare each a sum of two squares, then so is NM.

For example, 513 = (12,625)2+ (31,250)2.

See also

SQUARE

.

coefficient A numerical or constant multiplier of

the variables in a term of an algebraic expression is

called the coefficient of that term. For example, con-

sider the equation 5x3– 2x+ 7 = 0, where xis the

variable, the coefficient of x3is 5, the coefficient of x

is –2, and the coefficient of x2is zero. In the equation

3 cos y– 4xy2= 7, the coefficients of cos yand xy2

are 3 and –4, respectively.

Sometimes the value of a coefficient is not known

and a symbol is used in its stead. For instance, in the

expression ax2+ bx + cwith xthe variable, the num-

bers aand bare coefficients (and cis a constant term).

Although the values of a, b, and care not specified, it is

understood that their values do not change even as the

value of xvaries.

In a more general context, the term coefficient is

used for any number that serves as a measure of some

property or characteristic of a set of data or a physical

property. For instance, a

CORRELATION COEFFICIENT

in

STATISTICS

gives a measure of the extent to which two

data sets are interdependent, while the heat coefficient

in physics gives a measure as to how well a material

conducts heat.

See also

BINOMIAL COEFFICIENT

;

COMBINATO

-

RIAL COEFFICIENT

;

CONSTANT

;

LEADING COEFFICIENT

;

POLYNOMIAL

.

Collatz’s conjecture (“3n+ 1” mapping problem)

Consider the following process:

Select a positive integer. If it is odd, triple it

and add one; otherwise, divide the number by

two. Now perform the same operation again

on the result. Repeat this process indefinitely

to produce a sequence of numbers.

The number 7, for example, yields the sequence:

7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16,

8, 4, 2, 1, 4, 2, 1, 4, 2, 1,…

Notice that this sequence finally falls into a 4-2-1 cycle.

In 1937 German mathematician Lothar Collatz

conjectured that, no matter the starting integer selected,

all sequences lead to the same 4-2-1 cycle. Collatz was

unable to prove this claim, but he was also unable to

find an example of a starting number that does not

behave this way. To this day, no one knows whether or

not Collatz’s conjecture is true. All integers up to 2.702

×1016 have been checked.

As a first step toward understanding this problem,

mathematicians have proved that 4-2-1 is the only cycle

of reasonable size that could possibly appear; it has

been established that any other cycle that might appear

would be at least 275,000 numbers long.

collinear Any number of points are said to be

collinear if they all lie on the same straight line. Two

points are always collinear. Three points in a plane A=

(a1,a2), B= (b1,b2), and C= (c1,c2) are collinear only if

the lines connecting points Aand Band the points con-

necting Aand Chave the same

SLOPE

. This means that

the following relationship must hold: